
We provided a complete and self-contained derivation of the mode expansions for the spin-2 and spin-0 states and their interactions.
Generalizing the presentations in [1]}, [2]}, [3]}, our computations are given in de-Donder gauge for massless gravitons – allowing us to consistently compute scattering amplitudes involving intermediate off-shell states.

In previous work [4]} we had demonstrated the extended sum-rule relationships between spin-2 and spin-0 modes, and their masses and couplings, which must be satisfied in order for elastic massive spin-2 KK scattering to grow no faster than \({\cal O}(s)\) . Here, we have provided an analytic proof for one combination of these sum rules, and showed its relation both to the Einstein and scalar background field equations implementing the Goldberger-Wise dynamics and also to the properties of the mode-equations for the physical scalar fields (fields which are admixtures of bulk scalar and gravitational modes in the original theory).

We have provided, in an appendix to this work, a complete list of the sum-rule relations which must be satisfied if all \(2 \rightarrow 2\)  massive spin-2 scattering amplitudes, elastic or inelastic, are to grow no faster than \({\cal O}(s)\)  – completing the analyses begun in [5]}, [6]}, [7]}, [8]}.

Finally, using a version of the DFGK model [9]} in which the Goldberger-Wise dynamics can be treated perturbatively [4]}, we have checked numerically that the sum rules which enforce the proper high-energy behavior of massive spin-2 scattering continue to be satisfied in the case of the large warping that would be required to produce the hierarchy between the weak and Planck scales. These numerical computations demonstrate that, in models with a massive radion, proper cancellation is achieved only after including the contributions from the tower of scalar states present in the Goldberger-Wise model.

